The space of clouds in an Euclidean space

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The space of clouds in an Euclidean space

HAUSMANN, Jean-Claude, RODRIGUEZ, Eugenio

HAUSMANN, Jean-Claude, RODRIGUEZ, Eugenio. The space of clouds in an Euclidean space.

Experimental Mathematics, 2004, vol. 13, no. 1, p. 31-47

arxiv : math/0207107v1

Available at:

http://archive-ouverte.unige.ch/unige:12257

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arXiv:math/0207107v1 [math.DG] 12 Jul 2002

The space of clouds in an Euclidean space

Jean-Claude HAUSMANN and Eugenio RODRIGUEZ July 12 2002

Abstract

We study the spaceNd^m of clouds inR^d (ordered sets ofm points modulo the action of the group of affine isometries). We show thatN_d^mis a smooth space, stratified over a certain hyperplane arrangement inR^m. We give an algorithm to list all the chambers and other strata (this is independent of d). With the help of a computer, we obtain the list of all the chambers for m ≤ 9 and all the strata when m ≤ 8. As the strata are the prod- uct of a polygon spaces with a disk, this gives a classification of m-gon spaces for m ≤9. When d= 2,3, m = 5,6,7 and modulo reordering, we show that the chambers (and so the different generic polygon spaces) are distinguished by the ring structure of their mod 2-cohomology.

1 Introduction

Let E be an oriented finite-dimensional Euclidean space. Let NE^m be the space of ordered sets of m points in E, modulo the group of rigid motions of E; more precisely,

NE^m :=G(E)\E^m,

where the Lie group G(E) is the semi-direct product of the translation group of E by SO(E), the group of linear orientation-preserving isometries of E, and the groupG(E) acts diagonally onE^m. We shall occasionally consider the space N¯E^m = ¯G(E)\E^m, where ¯G is the group of all affine isometries of E^m. Observe that ¯NE^m is a subspace of NE^m′ whenE is a proper subspace ofE^′. An element of NE^m will be called a cloud of m points in E. (The letter N stands for “nuage”, meaning “cloud” in French.) We abbreviate N_R^md to Nd^m. Observe that NE^m is canonically homeomorphic toNd^m, when d= dimE.

The space Nd^m plays a natural role in celestial mechanics, at least for d = 2 or 3 (see, for instance, [AC]). Moreover, its importance was recognized espe- cially in statistical shape theory, a subject which has developed rapidly during the last two decades (see [Sm] and [KBCL]) for a history). There, the spaceNd^m

is called the size-and-shape space and is denoted by SΣ^m_d [KBCL, §11.2]. This

terminology and notation emphasizes thatNd^m is the cone, with vertex N0^m, over the shape space Σ^m_d, defined as the quotient of Nd^m− N0^m by the homotheties.

A great amount is known about the homotopy type of shape spaces. For in- stance, in [KBCL], Kendall, Barden, Carne and Li show that Σ^m_d admits cellular decompositions leading to a complete computation of its homology groups.

In this paper, we present an alternative decomposition of the spaceNd^m. It is based on polygon spaces, a subject which has also encountered a rich development during the last decade, in connection with Hamiltonian geometry. This approach is completely different from that of statistical shape theory and this paper is essentially self-contained.

First of all, the point set topology of Nd^m is well behaved and Nd^m is en- dowed with a smooth structure. More precisely, the translations act freely and properly on E^m with quotient diffeomorphic to the vector subspace K(E^m) = {(z1, . . . , zm) ∈ E^m | P

zi = 0}. Being therefore the quotient of K(E^m) by the action of the compact groupSO(E), the space of clouds NE^m is locally compact (in particular Hausdorff). Classical invariant theory provides a proper topological embeddingϕofNd^minto an Euclidean spaceR^N (see 2.2). This embedding makes NE^m a smooth space, i.e. a topological space together with an algebra C^∞(X) of smooth functions (with real values): those functions which are locally the com- position of ϕ with a C^∞-function on R^N. One can prove that f ∈ C^∞(NE^m) if and only if f^◦π is smooth on E^m, where π : E^m → NE^m is the natural projec- tion (Proposition 2.3). Any subspace of of a smooth space naturally inherits a smooth structure and, together withsmooth maps (see 2.1), smooth spaces form a category whose equivalences are called diffeomorphisms. Finally, we mention that the spaceN3^m has the special feature that the smooth maps admit a Poisson bracket (see 2.6).

Our main tool for stratifying the space NE^m is the mapℓ :NE^m →R^m defined onρ= (ρ1, . . . , ρm)∈E^m by

ℓ(ρ) := (|ρ₁−b(ρ)|, . . . ,|ρ_m−b(ρ)|), where b(ρ) = _m¹ P

ρi is the barycentre of ρ. This map ℓ is continuous and is smooth on ℓ⁻¹((R>0)^m), the open subset of points ρ ∈ NE^m such that no ρi is equal tob(ρ).

We shall prove in Section 3 that the critical points ofℓare the one-dimensional clouds ¯N1^m ⊂ NE^m. The space of critical values is then an arrangement of hyper- planes inR^m that we shall describe now. Let m :={1,2, . . . , m} and denote by P(m) the family of subsets ofm. For I ∈ P(m), let H^I be the hyperplane of R^m defined by

HI :={(a1, . . . , am)∈R^m |X

i∈I

ai =X

i /∈I

ai}.

We call these hyperplanes walls. They determine a stratification H(R^m) of R^m,

i.e. a filtration

{0}=H⁽⁰⁾(R^m)⊂ H⁽¹⁾(R^m)⊂ · · · ⊂ H^(m)(R^m) =R^m,

with H^(k)(R^m) being the subset of those a ∈ R^m which belong to at least m− k distinct walls HI. A stratum of dimension k is a connected component of H^(k)(R^m)− H^(k−1)(R^m). Note that a stratum of dimension k ≥ 1 is an open convex cone in ak-plane ofR^m. Strata of dimension m are called chambers. We denote by Str(a) the stratum of a∈R^m. If Str(a) is a chamber, the m-tuple ais calledgeneric and we will often denote Str(a) by Ch(a).

The stratification H(R^m) induces a stratification of any subset U of R^m, in particular ofU = (R_>0)^m. Write Str(U) for the set of all the strata of U. Denote by NE^m(a) the preimage ℓ⁻¹({a}) of a∈R^m. Strengthening results of [HK2], we shall prove the following theorem in Section 3.

Theorem A Leta∈(R>0)^m. Then there is a diffeomorphism fromℓ⁻¹(Str(a)) onto to NE^m(a)×Str(a) intertwining the map ℓ with the projection to Str(a).

In the proof of Theorem A, we actually construct, when Str(a) = Str(b), a diffeomorphism ψba : NE^m(a)−−→ N^≈ E^m(b). One has ψab =ψ_ba⁻¹ and ψaa = id. For α∈Str((R_>0)^m), we will sometimes use the notationNE^m(α) for any of the spaces NE^m(a) with a∈α. This is in fact ambiguous because, in general,ψca 6=ψcb◦ψba, so one cannot use the maps ψba to define an equivalence relation on ℓ⁻¹(Str(a)) giving the points of NE^m(α). However, ψ_ca is isotopic to ψ_cb◦ψ_ba, and so the homotopy invariants ofNE^m(α), for instance the elements of its cohomology ring, are well defined.

Theorem A may provide good local models for describing the evolution of a cloud. This is especially likely when dimE = 2,3, where, for generica, the spaces NE^m(a) and thus ℓ⁻¹(Str(a)) are smooth manifolds (see below).

Theorem A shows that NE^m is obtained by gluing together pieces of the form NE^m(α)×α for various α ∈Str((R>0)^m). From this point of view, the following questions are natural.

1. Describe the set of all strata of (R>0)^m, in particular the set of chambers.

This combinatorial problem does not depend on E.

2. Describe NE^m(α) for allα ∈Str((R>0)^m).

3. Describe how a stratum of H(NE^m) is attached to its bordering strata of lower dimension.

The main issue of this paper is to answer Question 1 and, partly Question 2 above. It is convenient to take advantage of the right action of the symmetric group Sym_m on NE^m and on R^m by permutation of the coordinates (to deal directly with the smooth spaceNE^m/Sym_m and get a corresponding statement of

Theorem A, see 2.5). This action permutes the strata ofH((R>0)^m), and NE^m(α) is diffeomorphic to NE^m(α^σ) for σ ∈ Sym_m. The map ℓ is equivariant and each a∈(R>0)^m has a unique representative in R^m_ր, where

R^m_ր :={(a1, . . . , am)∈R^m| 0< a1 ≤ · · · ≤am}.

Therefore, the set Str((R_>0)^m)/Sym_m is in bijection with the set Str(R^m_ր).

In Sections 4 and 5, we show how to obtain a complete list of the elements of Ch(R^m_ր) and Str(R^m_ր). For this, we first show that the set of inequalities defining a chamber α of R^m_ր can be recovered from some very concentrated information which we call thegenetic codeof α. Abstracting some properties of these genetic codes gives rise to the combinatorial notion of a virtual genetic code. We design an algorithm to find all virtual genetic codes, with the help of a computer (the program in C⁺⁺ is available at [HRWeb]). Deciding which virtual genetic code is the genetic code of a chamber (realizability) is essentially done using the simplex algorithm of linear programming. We thus obtain the list of all the chambers of R^m_ր, with the restrictionm ≤9 due to the computer’s limited capacities. The set Str(R^m−1_ր ) is determined using an injection of Str(R^m−1_ր ) into Ch(R^m_ր) (see §5).

The number of elements of these sets is

m 3 4 5 6 7 8 9

|Ch(R^m_ր)| ^{2 3 7 21 135 2470 175428}

|Str(R^m_ր)| ^{3 7 21 117 1506 62254 ?}

(1.1)

It turns out that, for m ≤8, all virtual genetic codes are realizable, but not for m = 9: only 175428 out of 319124 are realizable. The non-realizable ones might well be of interest (see Remark 7.13).

Our algorithms produce, in each chamberα, a distinguished elementamin(α)∈ R^m_ր with integral coordinates and with P

ai minimal. Several theoretical ques- tions about these elements amin(α) remain open (see §4).

To describe the spaces NE^m(a) (Question 2 above), we note that NE^m(a) =SO(E)\{ρ∈E^m |

m

X

i=1

ρi = 0 and |ρi|=ai}. The condition Pm

i=1ρi = 0 suggests the picture of a closed m-step piecewise- linear path in E, whose ith step has length ai. Therefore, the space NE^m(a) is often called them-gon space (inE) of typea(we could call it the space of clouds

“calibrated ata”). These polygon spaces have been studied in different notations, especially for dimE = 2 and 3 where, for generic a, they are manifolds: see, for instance, [Kl], [KM], [HK1], [HK2]. For dimE >3 or fora non-generic, see [Ka1]

and [Ka2].

The classification of the polygon spaces NE^m(a), for generica, was previously known when dimE = 2,3 and m ≤5 (see, for instance [HK1, §6]). The genetic

codes introduced in this paper extend this classification up to m = 9. In §6, we give handle-decomposition information about the 6-gon spaces ¯N2⁶ for the 21 chambers of R⁶_ր. This type of method could be applied to any space NE^m(a) for generica. In addition to these geometric descriptions, algorithms were previously found which compute cohomological invariants of the spacesN3^m(a), for example their Poincar´e polynomial ([Kl, Th. 2.2.4], [HK2, Cor. 4.3]). This enables us, in Section 7, to compute the Betti numbers of the spaces N3^m(α) for m ≤ 9.

Moreover, presentations of the cohomology ring of N3^m(α) for any coefficients were given in [HK2, Th. 6.4]. This permits us to compute some invariants of the ring H^∗(N3^m(α);F₂) and prove in 7.10 the following result:

Proposition B For 5 ≤ m ≤ 7, the spaces N3^m(α) (or N¯2^m(α)), for distinct chambersα of R^m_ր, have non-isomorphic mod 2-cohomology rings.

Here, the ring structure ofH^∗(N3^m;Z₂) is important: the Betti numbers alone do not distinguish the spaces. Interestingly enough, the virtual genetic codes which are not realizable also give rise to non-trivial graded rings. We do not know if these rings are cohomology rings of a space, or of a manifold (see Remark 7.13).

The paper is organized as follows. In Section 2, we set the background of the smooth structure on NE^m which is used in Section 3 to prove Theorem A. In Section 4, we introduce the genetic code of a chamber and show how to obtain the list of all chambers of R^m_ր for m ≤ 9. In Section 5 we study the injection Str(R^m−1_ր ) into Str(R^m_ր) and show how to obtain the list of all strata of R^m_ր for m≤ 8. Section 6 contains our information on the spacesN3^m(a) and ¯N2^m(a) for generica. Section 7 is devoted the cohomology invariants of the polygon spaces.

Finally, the results of Sections 6 and 7 are applied in Section 8 to the case of hexagon spaces.

Acknowledgments: Both authors thank the Swiss National Fund for Scien- tific Research for its support. We are indebted to R. Bacher for suggesting the cohomology invariants(α) of Proposition 7.8.

2 The smooth structure on N

2.1 Smooth spaces and maps. For X a topological space, denote by C⁰(X) the R-algebra of continuous functions on X with real values. If h : X → Y is a continuous map, denote byh^∗ :C⁰(Y)→ C⁰(X) the map h^∗(f) =f^◦h.

Let X be a subspace of R^N. A map f : X → R issmooth if, for each x ∈X there exists an open setU of R^N containing x and a C^∞ map F :U → R which coincides withf throughout U∩X (compare [Mi, §1]). The smooth maps onX constitute a subalgebraC^∞(X) of C⁰(X).

More generally, if ϕ : X →R^N is a topological embedding of a spaceX into

R^N one may consider the subalgebra C^∞(X) =ϕ^∗(C^∞(ϕ(X)). We call C^∞(X) a smooth structure onX and X (or rather the pair (X,C^∞(X))) a smooth space.

A continuous map h : X → Y between smooth spaces is called smooth if h^∗(C^∞(Y)) ⊂ C^∞(X). The map h is a diffeomorphism if and only if it is a homeomorphism and h and h⁻¹ are smooth. It is a smooth embedding if h^∗(C^∞(Y)) = C^∞(X). A smooth embedding is thus a diffeomorphism onto its image.

2.2 The smooth structure on NE^m. Letκ :E^m →E^m be the linear projection κ(z1, . . . , zm) = (z1−b(z), . . . , zm−b(z)),

where b(z) = _m¹ P

z_i is the barycentre of z. The image of κ is K(E^m) and its kernel is the diagonal ∆ inE^m.

The normal subgroup E in G(E) of translations acts freely and properly on E^m and the quotient space E\E^m is the same as the quotient vector space E/∆.

The projection κ descends to a linear isomorphism ¯κ : E\E^m−−→ K^≈ (E^m). The space NE^m is now the quotient of K(E^m) by the action of the compact group SO(E). Therefore NE^m is a locally compact Hausdorff space.

Consider the m² polynomial functions on E^m given by z 7→ hκ(zi), κ(zj)i, where h,i denotes the scalar product on E. Choose an orientation on E. The determinants|κ(z_i1),· · ·κ(z_ik)|with i₁ <· · ·< i_k(k = dimE) are another family of (^m_k) polynomial functions on E^m. All these functions areG(E)-invariant and produce a continuous map ϕ : NE^m →R^N with N =m²+ (^m_k). It is an exercise to prove thatϕ is injective and proper. As NE^m is locally compact, the mapϕ is a topological embedding of NE^m into R^N and its image is closed (for a family of inequalities definingϕ(NE^m) as a semi-algebraic set, see [PS]).

The embeddingϕ endowsNE^mwith a smooth structure. The following propo- sition identifies C^∞(NE^m) with the smooth functions on E^m which are G(E)- invariant.

Proposition 2.3 Let f ∈ C⁰(NE^m). The following are equivalent:

(A) f ∈ C^∞(NE^m).

(B) There is a global C^∞-function F :R^N →R such thatf =F^◦ϕ.

(C) The map f◦π : E^m → R is C^∞, where π : E^m → NE^m denotes the natural projection.

Proof: It is clear that (B) implies (A). Conversely, letf ∈ C^∞(NE^m). For every ρ ∈ NE^m, one has an open set Uρ of R^N containing ϕ(ρ) and a smooth function Fρ:Uρ→Rwithfρ◦ϕ =f onϕ⁻¹(Uρ). CallU∞ =R^N−ϕ(NE^m) andF∞:U∞→

R the constant map to 0. As ϕ(NE^m) is closed, the family U :={Uρ}ρ∈N_E^m∪{∞} is an open covering of R^N. Let µ_ρ : R^N → R be a smooth partition of the unity subordinated to U. Then F(x) = P

ρ∈N_E^m∪{∞}Fρ(x) satisfies (B).

Statement (B) is obviously stronger than (C) (which, incidentally, implies that π is a smooth map). For the converse, one uses that the components of ϕ con- stitute a generating set for the algebra of SO(E)-invariant polynomial functions onK(E^m) [Wl,§II.9]. Then, anySO(E)-invariant smooth function onK(E^m) is of the formF^◦ϕ by the Theorem of G. Schwarz [Sch].

2.4 The smooth structure on the space ¯NE^m =G(E)\E^m is obtained as in 2.2.

The embedding ϕ : ¯NE^m →R^m2 is given by the polynomial functionρ 7→ hρ_iρ_ji. Proposition 2.3 holds true.

2.5 Clouds of unordered points. On E^m ={ρ : m → E}, the symmetric group Sym_m acts on the right, by pre-composition (or by permuting the coordinates).

This action descends onNE^m.

As in 2.2, the space NE^m/Sym_m =G(E)\E^m/Sym_m has a smooth structure, via a topological embeddingϕ :R^m/Sym_m →R^N given by a generating set of the algebra of polynomial functions onK(E^m) which are SO(E)×Sym_m-invariant.

Proposition 2.3 holds true accordingly.

The space R^m/Sym_m also has a smooth structure via the smooth embedding ϕ : R^m/Sym_m → R^m given by the m elementary symmetric polynomials. The mapℓdescends to a continuous map ¯ℓ:NE^m/Sym_m→ R^m/Sym_mwhich is smooth away from ℓ⁻¹({0}). The composition ψ : R^m_ր ⊂ (R>0)^m → (R>0)^m/Sym_m is a smooth homeomorphism. The stratification H(R^m_ր) can be transported via ψ to (R_>0)^m/Sym_m, giving rise to a stratification H((R_>0)^m/Sym_m). The map ℓ¯is stratified and Theorem A holds true for ¯ℓ. Indeed, the diffeomorphisms constructed in the proof of Theorem A given in §3 are natural with respect to the action of Sym_m.

We must be careful that the smooth homeomorphismψ :R^m_ր →(R_>0)^m/Sym_m is not a diffeomorphism: the projection onto the first coordinate is smooth on R^m_ր but not on (R_>0)^m/Sym_m.

2.6 Poisson structures on N3^m. Recall that a Poisson structure on a smooth manifoldX is a Lie bracket {,}onC^∞(X) satisfying the Leibnitz rule: {f g, h}= f{g, h} + {f, h}g. See [MR] for properties of Poisson manifolds. The same definition makes sense on a smooth space.

The Euclidean space E =R³ has a standard smooth structure by {f, g}(x) := h∇f× ∇g, xi.

We endow the product space E^m with the product Poisson structure. If f, g : E^m →Rare SO(E)-invariant, so is the bracket {f, g}. Thus, the quotient space SO(E)\E^m inherits a Poisson structure.

Using a canonical identification of R³ with so(3)^∗, the above Poisson bracket onR³ corresponds, up to sign, to the classical Poisson structure on so(3)^∗ [MR, p. 287]. The mapµ:z 7→Pm

i=1zi, fromR³ =so(3)^∗ toE is the moment map for the diagonal action ofSO(E). Letξ :R³ →Rbe a linear map. By the Theorem of Noether [MR, Th. 11.4.1], if f : E^m → R is a smooth SO(3)-invariant map, then {f, ξ^◦µ} = 0. This proves that {f, g} = 0 for all g ∈ C^∞(E^m) such that g|K(E^m)= 0. Thus, the spaceN3^minherits a Poisson structure so that the inclusion N3^m ⊂SO(E)\E^m is a Poisson map.

When a ∈ (R>0)^m is generic, the spaces N3^m(a) are manifolds and are the symplectic leaves of ℓ⁻¹(Str(a)). This accounts for the symplectic structures on the polygon spaces inR³ studied in [Kl], [KM] and [HK1 and 2].

3 Proof of Theorem A

Throughout this section, the Euclidean space E and the number of points are constant. Denote by ˙K the subset of m-tuples ρ = (ρ1, . . . , ρm)∈ E^m such that ρi 6= 0 andPm

i=1ρi = 0. Define the map ˜ℓ: ˙K →R^m by ˜ℓ(ρ) := (|ρ1|, . . . ,|ρm|).

An element ρ= (ρ1, . . . , ρm) ∈ E^m is called 1-dimensional if the vector sub- space ofE spanned byρ1, . . . , ρm is of dimension 1 (therefore,ρrepresents an ele- ment of ¯N1^m ⊂ NE^m). These are precisely the singularities of the map ˜ℓ : ˙K →R^m. Indeed:

Lemma 3.1 Suppose that ρ∈K˙ is not 1-dimensional. Then Tρℓ˜is surjective.

Proof: Let (a1, . . . , am) = ˜ℓ(ρ). Asρis not 1-dimensional, there are two vectors among ρ2, . . . , ρm which are linearly independent. The orthogonal complements to these two vectors then span E. Thus, there are curves ρi(t) for i = 2, . . . , m such that|ρi(t)|=ai and

m

X

i=2

ρi(t) =−(1 + t a1

)ρ1. Therefore the map

t 7→((1 + t a1

)ρ1, ρ2(t), . . . , ρm(t))

represents a tangent vector v ∈ TρK˙ with Tρℓ(v) = (1,˜ 0, . . . ,0). The same can be done for the other basis vectors ofR^m proving thatTρℓ˜is surjective.

Letρ ∈K˙ be 1-dimensional. One thus has ρi =λiρm with λi ∈R− {0}. Let I(ρ) ∈ P(m) defined by i ∈ I(ρ) if and only if λi < 0. It is obvious that ˜ℓ(ρ) belongs to the wall HI(ρ).

Lemma 3.2 Suppose that ρ ∈ K˙ is 1-dimensional. Then the image of Tρℓ˜is HI(ρ).

Proof: Let I =I(ρ). One has P

i∈Iρi =−P

i /∈Iρi. The components ρi(t) of a curve ρ(t)∈E^m with ρ(0) =ρ are of the form

ρi(t) = (1 +ci(t) ai

)ρi+wi(t),

with c_i(0) = 0 and w_i(0) = 0, where c_i(t) ∈ R, and w_i(t) is in the orthogonal complement of ρi. The curveρ(t) is in ˙K if and only if Pm

i=1wi(t) = 0 and X

i∈I

(1 + ci(t)

a_i )ρi =−X

i /∈I

(1 + ci(t)

a_i )ρi. (3.2)

Letc(t) = (c1(t), . . . , cm(t)). The vector ^ρ_aⁱ

i is constant when i∈I and ^ρ_aⁱ

i =−^ρ_a^j_j if i∈I and j /∈ I. Therefore, Equation (3.2) is equivalent toc(t)∈ HI. Finally, a direct computation shows that the tangent vector v ∈TρK˙ represented byρ(t) satisfiesTρℓ(v) = ˙˜ c(0). This proves the lemma.

Proof of Theorem A : Let a, b ∈ (R_>0)^m be in the same stratum α. Let X⊂ αbe the segment joiningatob. Forδ >0, writeUδ :={x∈R^m |d(x, X)<

δ}, where d(x, X) is the distance from x to the segment X. We choose δ small enough so that the walls meeting Uδ, if any, are only those containing α. Let U˜δ := ˜ℓ⁻¹(Uδ)⊂K˙.

Let V^b be a vector field onUδ of the form V_x^b =λ(x)(b−a), where λ:Uδ → [0,1] is a smooth function equal to 1 on Uδ/3 and to 0 out ofU2δ/3.

Put on ˙K and R^m the standard Riemannian metrics. For ρ ∈ K˙, define the vector subspace ∆ρofTρK˙ by ∆ρ:= (Tρℓ)˜^♯(Tℓ(ρ)(R^m)), where (Tρℓ)˜^♯is the adjoint of Tρℓ. The vector spaces˜ δρ form a smooth distribution (of non-constant rank) on ˙K.

The tangent map Tρℓ˜sends ∆ρ isomorphically onto the image of Tρℓ. Since˜ X lies in α, Lemmas 3.1 and 3.2 show that V_ℓ(z)˜^b is in the image of Tzℓ˜for all z ∈ U˜_δ. Therefore, there exists a unique vector field W^b on ˜U_δ such that, for each z ∈ U˜δ, one has W_z^b ∈ ∆z and Tzℓ(W˜ _z^b) = V_˜ℓ(z)^b . The map ˜ℓ being proper, the vector fieldW^b has compact support, so its flow Φt is defined for all times t.

Therefore, z 7→Φ1(z) gives a diffeomorphism ψba : ˜ℓ⁻¹(b)−−→^≈ ℓ˜⁻¹(a).

As its notation suggests, the mapψba depends only onband not on the choices involved in the definition of V^b (δ and λ). One can thus define ψ : ˜ℓ⁻¹(α) → NE^m(a)×α by ψ(z) := (˜ℓ(z), ψℓ(z)a˜ (z)). The vector fields V^b and W^b depending smoothly on b ∈ α, the map is smooth as well as its inverse (x, u) 7→ ψax(u).

Therefore,ψ is a diffeomorphism. As the Riemannian metric on ˙Kand the map ˜ℓ are invariant with respect to the action ofSO(E)×Sym_m the mapψ descends to a diffeomorphismψ :ℓ⁻¹(α)−−→ N^≈ E^m(a)×α, which proves Theorem A. Actually, each diffeomorphism ψba descends to a diffeomorphism ψba :NE^m(b)−−→ N^≈ E^m(a).

Remark 3.3 Theorem A is also true for the spaces ¯NE^m.

4 The genetic code of a chamber

Leta∈(R≥0)^m. Following [HK2,§2], we define S(a)⊂ P(m) by I ∈S(a) ⇔ X

i∈I

ai ≤X

i /∈I

ai. (4.1)

The very definition of the stratification H implies that S(a) = S(a^′) if and only if Str(a) = Str(a^′). Thus, for α a stratum of (R>0)^m, we shall writeS(α) for the common setS(a) with a∈α.

When α is a chamber, the inequalities in (4.1) are all strict. The elements of S(α) are then, as in [HK2,§2], calledshort subsets of m. Observe thatA ∈m is short if and only if its complement ¯A is not short. Therefore, ifα is a chamber, the setS(α) contains 2^m−1 elements.

Define Sm(α) := S(α)∩ P^m(m), where P^m(m) :={X ∈ P(m)|m∈X}. Lemma 4.2 Let α∈Ch((R>0)^m). Then S(α) is determined by Sm(α).

Proof: One has

I ∈S(α) ⇐⇒







m ∈I and I ∈Sm(α) or

m /∈I and ¯I /∈Sm(α).

(4.2)

Let us now restrict ourselves to chambers ofR^m_ր. We shall determine them by a very concentrated information called their “genetic code”. Define a partial order

“֒→” on P(m) by saying that A ֒→ B if and only if there exits a non-decreasing mapϕ :A→B such thatϕ(x)≥x. For instanceX ֒→Y ifX ⊂Y since one can takeϕ being the inclusion. Thegenetic codeofαis the set of elements A1, . . . , Ak

ofS_m(α) which are maximal with respect to the order “֒→”. By Lemma 4.2, the chamberα is determined by its genetic code; we write α=hA1, . . . , Aki and call the sets Ai the genes of α. Thanks to (4.2), the explicit reconstruction of S(α) out of its genetic code is given by the following recipe.

Lemma 4.3 Let α=hA₁, . . . , A_ki ∈Ch(R^m_ր). Let I ∈ P(m). Then

I ∈S(α) ⇐⇒







m∈I and ∃j ∈k with I ֒→Aj

or

m /∈I and I¯6֒→Aj ∀j ∈k.

Example 4.4 To unburden the notations, a subset A of m is denoted by the number whose digits are the elements of A in decreasing order; example: 531 = {5,3,1}. InR³_ր, there are 2 chambers. One of them, sayα0, contains points such as (1,1,3) which are not in the image of ℓ :NE³ →R³. One has S3(α0) = ∅. Its genetic code is empty and one has

α0 =hi ; S(α0) ={∅,1,2,21}. The other,α1 contains (1,1,1), and one has

α1 =h3i ; S(α1) ={∅,1,2,3}.

Let us now figure out which subset A ⊂ Pm(m) is the genetic code of a chamber ofR^m_ր. To reduce the number of trials, observe that ifα=hA1, . . . , Aki, then

(a) A_i 6֒→A_j for all i6=j and (b) ¯Ai 6֒→Aj for all i, j.

Indeed, one has Condition (a) since the setsAi are maximal (and we do not write them twice). For Condition (b), if ¯Ai ֒→Aj, thenAi would be both short and not short and Inequalities (4.1) would have no solution. A finite set {A1, . . . , Ak}, with Ai ∈ Pm(m) satisfying Conditions (a) and (b) is called a virtual genetic code (of type m), and we keep writing it by hA1, . . . , Aki. Let G^m be the set of virtual genetic codes andG^m(k)the subset of those virtual genetic codes containing k genes.

The determination of Gm is algorithmic:

1. G^m(0) ={hi}.

2. Each A∈ Pm(m) satisfying ¯A6֒→Agives rise to a virtual genetic codehAi. This gives the setG^m(1).

3. Suppose, by induction, that we know the set G^m(k) Then, each

(hA1, . . . , Aki,hAk+1i) in G^m(k)× G^m(1), so that {A1, . . . , Ak+1} satisfies Con- ditions (a) and (b), gives rise to an element of G^m(k+1).

When G^(k+1) =∅, the process stops and Gm =Sm i=0G^m(k).

Examples 4.5 For m = 3, the family P3(3) contains the sets 3, 31, 32 and 321 (with the notations introduced in Example 4.4). Only 3 satisfies ¯3 = 21 6֒→ 3.

Thus G3⁽¹⁾ = {3} while G⁽²⁾ is empty. We deduce that G³ = {hi,h3i}. They correspond to the two chambers of R³_ր found in Example 4.4. In the same way, we easily find the following table:

m Elements of Gm

2 hi. 3 hi,h3i. 4 hi,h4i, h41i.

5 hi,h5i, h51i,h52i, h53i, h54i,h521i.

Having found the virtual genetic codes of type m, the next question is which of them are realizable, that is, which of them is the genetic code of a chamber of R^m_ր. We proceed as follows. Each virtual genetic codes hA1, . . . , Aki of type m determines, a subset ShA1,...,Aki by the recipe of Lemma 4.3. Define the open polyhedral coneP :=PhA1,...,Aki by

P :=n

x∈R^m_ր

X

i∈I

xi <X

i /∈I

xi ∀I ∈ShA1,...,Aki

o .

If there exits α ∈ Ch(R^m_ր) with α = hA1, . . . , Aki, then α = PhA1,...,Aki. The realization problem is thus equivalent to P being not empty. To find a point insideP, we “push” its walls and consider:

P1 :=n

x∈R^m_ր

X

i∈I

xi ≤X

i /∈I

xi−1∀I ∈ShA1,...,Aki

o⊂P (4.5)

As P is an open cone in R^m, then P is not empty if and only if P1 is not empty.

Indeed, if P is not empty, then ∅ 6= P ∩Z^m_ր ⊂ P1. We then use the simplex algorithm of linear programming to minimize the ℓ1-norm Pm

i=1xi on P1. This algorithm either outputs an optimal solution, which is a vertex ofP1, or concludes that P1 is empty [Ch].

A program in C⁺⁺ was designed, following the above algorithms (comments on this program and the source code can be found in [HRWeb]). A computer could thus list all the chambers ofR^m_ր for m ≤9. Each chamber α is given by a distinguished element amin(α)∈Z^m_ր with minimal Pm

i=1ai. The number of these chambers, |Ch(R^m_ր)| = |Ch((R>0)^m)/Sym_m| is the one given in the first line of Table (1.1) in the introduction.

Experimentally, it turned out that, for m ≤ 8, all virtual genetic codes are realizable. This is not true form = 9:

Lemma 4.6 The virtual genetic code h9642i ∈ G⁹ is not realizable.

Proof: Let S := Sh9642i. As 9531 ֒→ 9642, one has 9531 ∈ S. On the other hand, 9642 = 87531 ∈/ S. If S = S(a) for some generic a ∈ R⁹_ր, we would have a7+a8 > a9. Now 965 ∈/ S by Lemma 4.3, therefore 965 = 874321∈ S. By the above inequality on the ai’s, this would imply that 94321∈ S which contradicts 943216֒→9642.

Our algorithm found 319124 elements inG9, out of which 175428 are realizable.

The list of all the chambersα of R^m_ր with their representativeamin(α) can be found further in this paper form ≤ 6 (Sections 6 and 8) and on the WEB page [HRWeb] form = 7,8,9.

Several theoretical questions about amin(α) remain open. For example, why a_min(α) has integral coordinates (with the ℓ₁-norm |a|1 =P

a_i odd)? A priori, the vertices ofP1 should only be in Q^m_ր. Isamin(α) always unique? This suggests the following

4.7 Conjectures :

a) any stratum of α ∈ H(R^m_ր) contains a unique element a_min(α)∈ Z^m_ր with minimal ℓ1-norm.

b)α is a chamber if and only |amin(α)|1 is an odd integer.

c) All vertices of P1(S)have integral coordinates.

Conjecture b) is supported by the following evidences. First, it is obvious that an element a∈Z^m_ր with |a|1 odd is generic. On the other hand, it is exper- imentally true for m≤9. Conjecture a) for non generic strata is experimentally true for m≤8 (see Section 5). Conjecture c) has been checked form≤8.

4.8 Cuts: One can prove that the set Gm of virtual genetic code of type m is in bijection with the set of “cuts” on m (the name is given in analogy with the Dedeckind cuts of the rationals). A subset S of P(m) is acut if, for all I, J ⊂m the two following conditions are fulfilled:

(A) I ∈S ⇔I /¯∈S.

(B) if I ∈S and J ֒→I, then J ∈S.

The bijection sends a cutS ofm to the set of maximal elements (with respect to the order “֒→”) of S_m. For details, see [HRWeb].

5 Non generic strata

If a ∈ R^m_ր is not generic, some inequalities of (4.1) are equalities. Thus, an element I ∈ S(a) is either a short subset of m (strict inequality) or an almost short subset. As in Lemma 4.2,S(a) is determined bySm(a) =S(a)∩Pm(m) and the latter is determined by those elements which are maximal with respect to the order “֒→” (the genes of S(a)). We denote the genes which are short subsets by A1, . . . , Ak and those which are almost short byB₁⁼, . . . , B_l⁼. For instance, when m = 3, one writes S(1,1,1) = h3i and S(1,1,2) = h3⁼i. To be more precise on our conventions, let I⁼ be an almost short gene of S(a) and J ֒→I. If |J|<|I|, then J is automatically short (since a ∈ R^m_ր ⊂ (R_>0)^m). If |J| = |I|, then J is supposed to be almost short unless there is a short gene K with J ֒→ K. For instance, S(1,2,2,3,4) =h51,53⁼i.

The set Str(R^m_ր) of all the strata of H(R^m_ր) will be studied via a mapα7→α⁺ from Str(R^m−1_ր ) to Ch(R^m_ր) which we define now. Let a ∈ R^m−1_ր . If ε is small enough, the m-tuple a⁺ := (δ, a1, . . . , am−1) is a generic element of R^m_ր for δ < ε and α⁺ := Ch(a⁺) depends only on α= Str(a).

If β = α⁺, we denote α = β⁻. This makes sense because of the following lemma.

Lemma 5.1 The map α7→α⁺ is injective.

Proof: Let a, b∈R^m−1_ր such that Str(a)6= Str(b). The segment joining a to b will then cross a wall H^I which does not contain Str(a). But then, the segment joining a⁺ and b⁺ will also cross HI, showing that Ch(a⁺)6= Ch(b⁺).

The correspondence α 7→ α⁺ can easily be described on the genetic codes.

The genetic code of a⁺ has the same number of genes than that of a. The correspondence goes as follows. If {p1. . . pr} is a gene of S(a) which is short, then {p⁺₁ . . . p⁺_r,1}is a gene of S(a)⁺, where p⁺_i =pi+ 1 (the genes of S(a⁺) are all short). If {p₁. . . p_r}⁼ is an almost short gene of S(a), then {p⁺₁ . . . p⁺_r} is a gene of S(a⁺).

The following convention will be useful.

5.2 Let a = (a1, . . . , ak) be a generic element of Z^k_ր. For m ≥ k, the m-tuple ˆ

a= (0, . . . ,0, a1, . . . , ak) determines a chamber ˆα represented by (δ1, . . . δm−k, a1, . . . , ak), where δi > 0 and P

δi <1. We say that ˆa is a conven- tional representative of ˆα. For instance, h521i having the conventional represen- tative (0,0,1,1,1) shows that h521i=h321i⁺⁺.

Lemma 5.3 Let α be a chamber of R^m_ր. Then α = β⁺ if and only if one (at least) of the two following statement holds:

1. α has a conventional representative (0, a₂, . . . , a_m).

2. there exists (a1, . . . , am)∈α∩Z^m with P

ai odd and a1 = 1.

Proof: It is clear that either 1. or 2. implies α = Str(a2, . . . , am)⁺. Also, if α=β⁺for β generic, then αadmits a conventional representative. It remains to show that, ifα=β⁺ with β non-generic, then Statement 2. holds true.

Observe that, as the walls H^I are defined by linear equations with integral coefficients, thenβ∩Q^m−1 is dense inβ. Asβ is a cone, it must contain a point in b ∈ Z^m−1_ր . As b is not generic, then P

b_i must be even and the m-tuples (δ, b1, . . . , bm−1) are all in α for δ <2.

Tables III-V of Section 6 and Table VI of Section 8 show thatamin(α) satisfies the above conditions for all α∈Ch(R^m_ր) when m≤6. This proves the following

Proposition 5.4 The correspondence α 7→ α⁺ gives a bijection Str(R^m−1_ր ) → Ch(R^m_ր) for m≤6.

Tables I and II below make the bijectionα 7→α⁻ from Ch(R^m_ր) to Str(R^m−1_ր ) explicit (we put a conventionala_min(α) when there exists one).

The bijection Ch(R⁴_ր)−−→^≈ Str(R³_ր)

α a_min(α) N2⁴(α) α⁻ a_min(α⁻) N2³(α⁻) hi (0,0,0,1) ∅ hi (0,0,1) ∅ h4i (1,1,1,2) S¹ h3⁼i (1,1,2) 1 point

h41i (0,1,1,1) S^1`S¹ h3i (1,1,1) 2 point

The bijection Ch(R⁵_ր)−−→^≈ Str(R⁴_ր)

α amin(α) N2⁵(α) α⁻ amin(α⁻) N2⁴(α⁻) hi (0,0,0,0,1) ∅ hi (0,0,0, ,1) ∅ h5i (1,1,1,1,3) S² h4⁼i (1,1,1,3) 1 point

h51i (0,1,1,1,2) T² h4i (1,1,1,2) S¹ h52i (1,1,2,2,3) Σ^or₂ h41⁼i (1,2,2,3) S¹∨S¹ h521i (0,0,1,1,1) T^{2`</sup>T<sup>2</sup> h41i (0,1,1,1) S<sup>1`}S¹

h53i (1,1,1,2,2) Σ^or₃ h42⁼i (1,1,2,2)

r r

h54i (1,1,1,1,1) Σ^or₄ h43⁼i (1,1,1,1)

r

r r

Here, Σ^or_g stands for the orientable surface of genus g and the two graphs in the last column are the 2 and 3-fold covers ofS¹∨S¹ without loops. The same work with the bijection Ch(R⁶_ր)−−→^≈ Str(R⁵_ր) gives the classification of all the 21 pentagon spaces (not necessarily generic) obtained by A. Wenger [We].

In the above two tables, one sees that N2⁵(α) is the boundary of a regular neighborhood (here in R³) of N2⁴(α⁻). This reflects the following fact. Let a0 ∈ α without zero coordinate. For any a ∈ α, the Riemannian manifold NE^m(a) is canonically diffeomorphic toNE^m(a0) by Theorem A and its proof. This produces a family of Riemannian metricsgaonNE^m(a0), indexed bya∈α. When a tends to a point a⁻ ∈ α⁻, the Riemannian manifold (NE^m(a0), ga) converges, for the Gromov-Hausdorff metric, to the metric space NE^m−1(a⁻).

On the other hand, the map α 7→ α⁺ is not surjective when m ≥ 7. For instance, h764i, with amin = (2,2,2,2,3,3,3), is not of the form (α⁻)⁺. For, α⁻ would be h653⁼i. As 421 = 653 ֒→ 653, this would imply that all a⁻_i are equal and α⁻ =h654⁼i. But h654⁼i⁺=h765i 6=h764i.

The table of Ch(R⁷_ր) (giving the 135 7-gon spaces) shows 18 chambers with the first coordinateamin not equal to 0 or 1 (see [HRWeb]). One might ask whether

there are otherm-tuplesain these chambers witha1 = 0,1. But, by applying the simplex algorithm to minimize a₁ on the polytope P₁ of (4.5), we saw that this is not the case. Therefore |Str(R⁶_ր)| = 118. The same procedure succeeded for m= 8 and 9, giving the cardinality of Str((R>0)^m)/Sym_m = Str(R^m_ր) for m ≤8 listed in the introduction.

6 Geometric descriptions of the N

(α)’s

When d = 2 or 3 and a is generic, the spaces Nd^m(a) are smooth manifolds, since SO(d) acts freely on the non-lined configurations. The space ¯N2^m(a) is also a manifold and the map N2^m(a) → N¯2^m(a) is a 2-sheeted covering. The space ¯N2^m(a) lies in N3^m(a) as the fixed point set for the involution τ on N3^m(a) obtained by reflection through a hyperplane. Observe that dimN3^m = 2(m−3) while dim dim ¯N2^m = m−3. The manifold ¯N2^m plays the role of a real locus of N3^m(a), the latter being endowed with a natural Kaehler structure for which the involution τ is antiholomorphic (see [HK2, §9]). It is shown in [HK2, Th. 9.1]

that the cohomology rings H^2∗(N3^m(α);Z₂) and H^∗( ¯N2^m(α);Z₂) are isomorphic, by a graded ring isomorphism dividing the degrees by 2.

The above polygon spaces were previously known for m ≤ 5 (see, for in- stance, [HK1,§6]). Our classification by genetic code produces the more system- atic tables below. Conventional representatives amin(α) (see 5.2) are used when available.

Table III : the 3-gon spaces α amin(α) N3³(α) N¯2³(α) N2³(a) hi (0,0,1) ∅ ∅ ∅ h3i (1,1,1) 1 point 1 point 2 points

Table IV : the 4-gon spaces α amin(α) N3⁴(α) N¯2⁴(α) N2⁴(a) hi (0,0,0,1) ∅ ∅ ∅ h4i (1,1,1,2) CP¹ RP¹ S¹ h41i (1,2,2,2) S² S¹ S^1`S¹

Table V : the 5-gon spaces

α a_min(α) N3⁵(α) N¯2⁵(α) N2⁵(α)

1 hi (0,0,0,0,1) ∅ ∅ ∅

2 h5i (1,1,1,1,3) CP² RP² S² 3 h51i (0,1,1,1,2) CP²♯CP² Σ1 T² 4 h52i (1,1,2,2,3) (S²×S²)♯CP² Σ₂ Σ^or₂ 5 h521i (0,0,1,1,1) S²×S² T² T^2`T² 6 h53i (1,1,1,2,2) CP²♯3CP² Σ3 Σ^or₃ 7 h54i (1,1,1,1,1) CP²♯4CP² Σ4 Σ^or₄

Our method produces a classification of the spaces NR^mn(α) for m ≤ 9, α a chamber, andn ≥2. Table VI of Section 8 gives the list of hexagon spaces. The tables for generic m-gon spaces when m = 7,8,9 are too big to be included in this paper. They can be consulted on the WEB page [HRWeb].

We shall now give procedures describingNE^m(β⁺) in terms ofNE^m−1(β) when βis generic and dimE = 2,3. Am-tuple (ρ₁, . . . , ρ_m)∈ K(E^m) is called avertical configurationif ρm = (0, . . . ,0,−|ρm|).

Proposition 6.1 Ifβ ∈Ch(R^m−1_ր ), thenN2^m(β⁺)is diffeomorphic toN2^m−1(β)× S¹.

Proof: Let (b2, . . . , bm) ∈ β and let ε > 0 small enough so that a :=

(ε, b2, . . . , bm)∈β⁺. A class inN2^m(a) has a unique representativeρ= (ρ1, . . . , ρm) which is a vertical configuration. As b is generic, if ε is small enough, then (b2, . . . , b^′_m) ∈ β when |b^′_m −bm| < ε. The (m−1)-tuple (ρ2, . . . , ρm +ρ1) thus represents an element ρ^′ ∈ N2^m−1(β) and the correspondence ρ 7→ (ρ^′, ρ1) pro- duces a diffeomorphism fromN2^m(β⁺) to N2^m−1(β)×S¹.

Let ρ = (ρ1, . . . , ρm) ∈ (R³)^m. Let ρ^⊥_m be the orthogonal complement of ρm, oriented by the vector ρ. Let π : R³ → R² be the composition of the orthogonal projectionR³ →ρ^⊥_m with some chosen isometryρ^⊥_m −−→^≈ R² preserving the orientation. Ifa∈(R>0)^m is generic, thanπ(ρ1),π(ρ1+ρ2),. . . ,π(ρ1+· · ·+ ρm−2) are not all zero. This defines a smooth map

r:N3^m(α)→(R²)^m−2−{0}/SO(2) (6.1) where α = Ch(a). The right hand member of Equation (6.1) is homotopy equivalent to CP^m−3. The map r thus determines a cohomology class R ∈ H²(N3^m(α);Z) which is the characteristic class of some principal circle bundle E(α)→ N3^m(α). The class R was introduced in [HK2,§6 and 7] and will appear again in Section 6 below.

Lemma 6.2 (compare [HK2, Prop. 7.3]) The total spaceE(α)isS¹-equivariantly diffeomorphic to the spaces of representatives of N3^m(a), (Ch(a) = α) which are vertical configurations.

Proof: Let E^′(α) ⊂ (R³)^m be the space described in the statement. Any element ofN3^m(a) has at least one representative which is a vertical configuration and any two of those are in the same orbit under the orthogonal action of S¹ = SO(2) fixing the vertical axis. Asais generic, the quotient mapE^′(α)→ N3^m(a) is then a principal circle bundle. Ifπ :R³ →R²denotes the projection onto the first two coordinates, the correspondenceρ7→(π(ρ1), π(ρ1+ρ2), . . . , π(ρ1+· · ·+ρm−2)) defines a smoothS¹-equivariant map ˜r:E^′(α)→(R²)^m−2which covers the mapr.

This proves that the characteristic class ofE^′(α)→ N3^m(a) isR.

Example 6.3 The chamber α =hmi of R^m_ր has its minimal representative a = a_min(α) = (1, . . . ,1, m−2). As, in a vertical configuration ρ of [ρ] ∈ N3^m(a), one has Pm−1

i=1 ρi = (0, . . . ,0, m−2), the sequence of the third coordinate of ρ1, ρ1+ρ2, . . . , must be strictly increasing. This implies that the map ˜rof the proof of 6.2 is a smoothS¹-equivariant embedding. It induces a diffeomorphism from N3^m(hmi) onto CP^m−3 and an identification of the bundle E(hmi) → N3^m(hmi) with the Hopf bundle. (see also [Ha, Remark 4.2]).

LetD(α) be the total space of the D²-bundle associated toE(α)→ N3^m(α).

Proposition 6.4 If β ∈Ch(R^m−1_ր ), then N3^m(β⁺) is diffeomorphic to the double of D(β).

Proof: Let (b2, . . . , bm) ∈ β and let ε > 0 be small enough so that a :=

(ε, b2, . . . , bm) ∈ β⁺. A class in N3^m(a) has a representative ρ = (ρ1, . . . , ρm) which is a vertical configuration and with ρ1 = (εcosθ,0, εsinθ). Let ˇE(a) be the space of such representatives. The map sending ρ to θ is a smooth map θ: ˇE(a)→[0, π].

If ρ ∈ Eˇ(a), then ρ_m +ρ₁ is close to ρ_m. This defines a smooth map P : Eˇ(a)→ SO(3), sending ρtoPρ, characterized byPρ(ρm+ρ1) = (0,0,−|ρm+ρ1|) and Pρ= id if θ(ρ) = 0, π. The smooth map ˇF : ˇE(a)→[0, π]× E(β) given by

Fˇ(ρ) :=

θ(ρ), (Pρ(ρ2), . . . , Pρ(ρm−1), Pρ(ρm+ρ1)) is a diffeomorphism. It induces a diffeomorphism

F :N3^m(α)∼=N3^m(a)−−→[0, π]× E(β).

∼ (6.4)

where∼is the equivalence relation given by (0, η)∼(0, g·η) and (π, η)∼(π, g·η) for all g ∈ SO(2). The right member of (6.4) is diffeomorphic to the double of D(β) which proves the proposition.